Simple formula for integration of polynomials on a simplex

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We show that integrating a polynomial f of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous related Bombieri polynomials of degree $$j=1,2,\ldots ,t$$ , each at a unique point $$\varvec{\xi }_j$$ of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such integrals must be evaluated. A similar result also holds for a certain class of positively homogeneous functions that are integrable on the canonical simplex.